Intel Edison has 1.8V and 3.3V internal voltage regulators. So can we disable those regulators in software to save power (We are using the default Yocto project inside it Linux Edison 3.10.17-yocto-standard)?

Your diagram is incomplete. You assume that there are multiple voltage regulators: like two 1.8V regulators, the first one for the module itself, the second one for external devices. But it's not true.